Need Suggestions for Random Numbers

[Matthew J Ghio]

jpinson at fcdarwin.org.ec writes:

> I am working on a PC implementation of a one-time pad cipher, and
> am trying to develop a way to produce an "acceptable" random file
> to be xor'd with the plaintext.

My favorite is the fibbonachi (sp?) series.  You've probably seen this
before: The series begins with the first two numbers being ones, and
each number after if being the sum of the two preceeding numbers. 
Therefore, we have:

1,1,2,3,5,8,13,21,34,55,89,144,233...

Taking modulo 10, we get:

1,1,2,3,5,8,3,1,4,5,9,4,3,7,0,7,7,4,1,5,6,1,7,8,5,3,8,1,9,0,9,9,8...

Which gives a fairly random distribution of numbers from 0 to 9.  You
can take a different mod value to adjust the range of numbers produced. 
This will eventually repeat (in the mod 10 example example I believe it
will repeat after around 60 numbers - you'll never be able to get all
possible combinations, for example 0,0 is not possible) but the
distribution is fairly random.  Increasing the randomness, (and the
legnth before it will repeat) is easy.  For example if you make the
series the sum of the first two of the last five numbers you get (modulo
10 for simplicity):

1,1,1,1,1,2,2,2,2,3,4,4,4,5,7,8,8,9,2,5,6,7,1,7,1,3,8...

Although this starts off slowly, the randomness picks up, and this will
generate a series which will go for thousands of digits without
repeating.  By the way, I don't reccomend adding more than two numbers
together to get the next number in the series.  If you try adding three,
four, or more numbers together, it causes the series to increase faster,
which causes it to reach the point where it repeats sooner, plus it
complicates your software and slows down the computation.

Anyway, if after extending the series, it's still not random enough, try
this: Change your program so that after it adds the first two numbers,
it looks at the third number.  If this third number is greater, less
than, or equal to some arbritrary value, add the fourth number to the
first two and then uses that as the next digit in the series.  This will
greatly increase the random effect.  This makes an excellent cipher, as
you can generate different series based on what substitutions you make
in the series.  Of course, your ideas about randomizing further by
combining random noise files is good, just be careful when using xor,
because you could end up cancelling out the beginnings of your serieses,
(since all these series begin with 1,1, xoring them would give you
zeros.)  Of course also try changing the initial conditions of the
fibbonachi series, just be sure you don't use something that will lock
the series (such as 5,5 which will produce 5,5,0,5,5,0,5,5,0...) 
Re-encrypting the noise file is also a good idea, multiplying each byte
by three and then doing a mod 256 works well for these purposes.

> (I remember some years back someone demonstrated the Apple II
> random number generator was flawed by converting the random
> numbers to screen coordinates and "painting" the screen.  No
> matter how long you ran the program, certain areas of the screen
> were never filled in. In other words, certain numbers were never
> generated.)

Well, I've programmed on Apple II computers for years, and there were
two very common systems used for random number generation.  Applesoft
Basic simply read bytes in the ROM and used them as random numbers. 
6502 code looks pretty random when you're just looking at the numeric
opcodes and data.  The other popular thing to do was to read the video
count.  This works best when your program is interacting with a human,
because people don't always respond to prompts in exactly the same
amount of time every time, so the position the video circuitry was
scanning would be different almost every time the program was run.  This
method works best for providing a seed for a series generator like the
ones described above.  If your computer has a clock, just read the time,
and that will have the same random effect.